Mathovia

Math Practice

Rational Inequalities Practice Problems

Twelve hand-built rational inequalities with full worked solutions. Each gets one side to zero, combines into a single fraction, factors, tests each interval, and reports the answer — with denominator zeros always excluded.

  1. 1.

    Solve \(\dfrac{x - 2}{x + 3} \ge 0\).

    Show full solution

    Critical values \(2\) (num, include), \(-3\) (den, exclude). Test: \(+\), \(-\), \(+\).

    Answer: \((-\infty, -3) \cup [2, \infty)\).

  2. 2.

    Solve \(\dfrac{x + 1}{x - 4} \le 0\).

    Show full solution

    Critical values \(-1\) (num, include), \(4\) (den, exclude). Negative between.

    Answer: \([-1, 4)\).

  3. 3.

    Solve \(\dfrac{3}{x - 5} > 0\).

    Show full solution

    Numerator always positive; the fraction is positive where the denominator is positive.

    Answer: \((5, \infty)\).

  4. 4.

    Solve \(\dfrac{x - 6}{x + 1} < 0\).

    Show full solution

    Critical values \(-1, 6\). Negative between.

    Answer: \((-1, 6)\).

  5. 5.

    Solve \(\dfrac{x}{x + 2} \le 0\).

    Show full solution

    Critical values \(0\) (num, include), \(-2\) (den, exclude). Negative between.

    Answer: \((-2, 0]\).

  6. 6.

    Solve \(\dfrac{x}{x - 3} > 2\).

    Show full solution

    \(\dfrac{x}{x - 3} - 2 > 0 \Rightarrow \dfrac{x - 2(x - 3)}{x - 3} = \dfrac{-x + 6}{x - 3} > 0\). Critical values \(6\) (num), \(3\) (den). Positive on \((3, 6)\).

    Answer: \((3, 6)\).

  7. 7.

    Solve \(\dfrac{x + 4}{x - 1} \ge 2\).

    Show full solution

    \(\dfrac{x + 4}{x - 1} - 2 \ge 0 \Rightarrow \dfrac{x + 4 - 2(x - 1)}{x - 1} = \dfrac{-x + 6}{x - 1} \ge 0\). Critical values \(6\) (num, include), \(1\) (den, exclude). Positive on \((1, 6)\).

    Answer: \((1, 6]\).

  8. 8.

    Solve \(\dfrac{4}{x + 2} \le 1\).

    Show full solution

    \(\dfrac{4}{x + 2} - 1 \le 0 \Rightarrow \dfrac{2 - x}{x + 2} \le 0\). Critical values \(2\) (num, include), \(-2\) (den, exclude). Test: \(-\), \(+\), \(-\).

    Answer: \((-\infty, -2) \cup [2, \infty)\).

  9. 9.

    Solve \(\dfrac{x^{2} - 1}{x + 3} \le 0\).

    Show full solution

    \(\dfrac{(x - 1)(x + 1)}{x + 3} \le 0\). Critical values \(-3\) (den, exclude), \(-1, 1\) (num, include). Test: \(-\), \(+\), \(-\), \(+\).

    Answer: \((-\infty, -3) \cup [-1, 1]\).

  10. 10.

    Solve \(\dfrac{(x + 2)^{2}}{x - 3} \ge 0\).

    Show full solution

    Critical values \(-2\) (num, from squared factor), \(3\) (den, exclude). Sign: \(-\) on \((-\infty, -2)\), \(-\) on \((-2, 3)\), \(+\) on \((3, \infty)\) — no flip at \(-2\).

    Want \(\ge 0\): \((3, \infty)\), plus \(x = -2\) (numerator zero).

    Answer: \(\{-2\} \cup (3, \infty)\).

  11. 11.

    Solve \(\dfrac{x^{2} + x - 6}{x - 1} \le 0\).

    Show full solution

    \(\dfrac{(x + 3)(x - 2)}{x - 1} \le 0\). Critical values \(-3, 2\) (num, include), \(1\) (den, exclude). Test: \(-\), \(+\), \(-\), \(+\).

    Answer: \((-\infty, -3] \cup (1, 2]\).

  12. 12.

    Solve \(\dfrac{5}{x - 2} \ge 1\).

    Show full solution

    \(\dfrac{5}{x - 2} - 1 \ge 0 \Rightarrow \dfrac{7 - x}{x - 2} \ge 0\). Critical values \(7\) (num, include), \(2\) (den, exclude). Positive on \((2, 7)\).

    Answer: \((2, 7]\).
M
Written by
Mathovia Team
Editorial Team

Rational Inequalities Practice Problems

The Rational Inequalities lesson covers the sign-chart method with the two adjustments for fractions; this set is 12 to practice on, each fully solved.

How to use this set

These problems are hand-written, not auto-generated. Get one side to zero, combine into a single fraction, factor, list all critical values (numerator and denominator), test each interval, and select the matching intervals — including numerator zeros for / , never denominator zeros. Then compare with the solution.

Common mistakes to watch for

Cross-multiplying or clearing the denominator — its sign is unknown.

Including a denominator zero — never, even for / .

Forgetting the denominator’s zeros as critical values.

Leaving the inequality as “fraction vs. a nonzero number” instead of reducing to “vs. zero”.

Where to go next

The sign-chart mechanics are the same as the Polynomial Inequalities Practice Problems. Combining fractions is the Rational Expressions Practice Problems skill. The answer format is Interval Notation. For the full method, see the Rational Inequalities lesson.

Frequently Asked Questions

Why can't I multiply both sides by the denominator?+

The denominator's sign is unknown — sometimes positive, sometimes negative — and multiplying an inequality by a negative flips it. Instead, move everything to one side, combine into one fraction, and analyze its sign.

What are the critical values?+

Values that make the numerator zero (where the fraction can be zero) and values that make the denominator zero (where it is undefined). Both split the number line.

Are denominator zeros ever in the solution?+

Never — the expression is undefined there. Even for or , use a parenthesis at those points.

How do I handle a non-strict inequality's numerator zeros?+

Include them. For , a numerator zero makes the fraction zero, which satisfies .

Why combine into a single fraction first?+

An inequality like can't be sign-charted as written. Subtract 2 and combine to get a single fraction versus zero.

Why is this a curated set?+

The answers are intervals and unions, not single numbers. Each problem has a full worked solution with the sign table.

More practice problems

Polynomial Inequalities Practice Problems Rational Expression Practice Problems: Free Practice Evaluating Rational Expressions Linear Inequalities Practice Problems